Invited talk delivered at
the Fifth International Conference on Computing Anticipatory Systems:
(CASYS'01), Liége, Belgium, 13-18 August 2001. To appear in the American
Institute of Physics Conference Proceedings.

Abstract. So-called
hidden variables introduced in quantum mechanics by Louis de Broglie and David
Bohm have been revived in the recent works by the author. However, the
variables, as such, have changed their initial enigmatic meanings and acquired
quite reasonable outlines of real and measurable characteristics. The success
in the deepest description of quantum systems becomes possible due to the
detailed consideration of a background that directly or indirectly influences
the behavior of the quantum system studied. Namely, the start viewpoint was
the following: All the phenomena, which we observe in the quantum world,
should reflect structural properties of the real space. Thus the scale 10-28
cm at which three fundamental interactions (electromagnetic, weak, and
strong) intersect has been treated as the size of a building block of the
space. Hence, it turns out that our space looks like the topological
tessellation of a mathematical space with elementary balls (or cells, or
superparticles). The appearance of a massive particle is associated with a
local deformation of the cellular space, i.e. deformation of a cell. The
mechanics of a moving particle that has been constructed is deterministic by
its nature and shows that the particle interacts with cells of the space
creating elementary excitations (called "inertons") in a cellular
substrate. The further study has disclosed that inertons are a substructure of
the matter waves which are described by the wave ψ-function formalism. It
has been found that the range covered by the inerton cloud surrounding a
moving particle is defined by the relationship Λ=λc/v where λ
is the de Broglie wavelength, c is the speed of light and v is the velocity of
the particle (so just Λ limits the range of action of the ψ-function
formalism). The kinetics of a particle constructed in the cellular space
easily results in the Schrцdinger and Dirac formalisms. Besides, the
concept of the cellular elastic space has allowed resolving the spin problem,
which has been reduced to a special intrinsic particle oscillation. The
theory, or more exactly, the existence of inertons, has been verified
experimentally: in rarefied gases they are inerton clouds of atoms' electrons
which interact with a strong laser pulse and in a solid, atom's inertons
induce an additional harmonic potential that contributes to the interatomic
interaction (metal specimens and the KIO3.HIO3 crystal).

The main original
physical parameters of quantum theory are Planck's constant h and de Broglie's
wavelength λ. These two enter into the two major quantum mechanical
relationships for a particle proposed by Louis de Broglie (see, e.g. de
Broglie, 1986)

E = h ν; λ = h
/ p. (1)

Here p was the momentum
of the particle, ν was the peculiar particle's frequency that coincided
with the frequency of a wave that specified by the wavelength λ and
traveled together with the particle. Later when Schrödinger's equation
appeared and Heisenberg proposed the uncertainty relations, the interpretation
of the said characteristics changed. Namely, the notion of the particle was
transformed to a "particle-wave" and hence λ and ν became
characteristics of the particle-wave. Born interpreted the square of the
absolute magnitude of the wave ψ-function of the Schrödinger equation as
the probability of particle location in a place described by the radius vector
r. Thus Born finally rejected any physical interpretation from a set of
parameters that described a quantum system. Since the end of the 1920s, only
one parameter has been perceived as pure physical – the particle-wave
wavelength λ called the de Broglie wavelength. In experimental physics,
those waves received also another name – the matter waves. Such a name
directly says that corpuscles (but not dim "particle-waves") are
able to manifest a wave behavior.

Since 1952 de Broglie
followed two papers by Bohm (1952) (see also Bohm, 1996) turned back to his
initial ideas on the foundations of the wave mechanics of particles. De
Broglie (see, e.g. de Broglie, 1960, 1987) believed that a submicroscopic
medium interfered in the motion of a particle and the appropriated wave guided
the particle. He believed firmly the causal interpretation of quantum
mechanics and warned that the resolution of the issue should not be based on
the wave ψ-function formalism, as the ψ-function was determined only
in the phase space but not in a real one. His own attempts were aimed at
seeking for the form of the so-called double solution.

In the case of the Dirac
formalism things get worse. The formalism introduced new additional notions
such as spinors and Dirac's four-row matrices, which allowed the calculation
of the energy states of the quantum system studied and changes in the states
due to the influence of outside factors. However, the formalism did not
propose any idea on the reasons of the wave behavior of matter and a nature of
the particle spin.

So far, modern studies
devoted to the foundations of quantum mechanics have tried to reach the
deepest understanding of quantum theory reasoned that just the ψ-function
formalism is original and it is often exploit even on the scale of Planck
length Ö {Għ/c3}~ 10-33 cm. This is
especially true for quantum field theory including quantum gravity (see, e.g.
Wallace, 2000; Sahni and Wang, 2000). Besides, there are views that a
gravitationally induced modification to the de Broglie's wave-particle duality
is needed when gravitational effects are incorporated into the quantum
measurement process (Ahluwalia, 1994, 2000; Kempf et al., 1995). Other
approaches try to introduce a phenomenological description based on the metric
tensor gij in typical quantum problems ('t Hooft, 1998). Classical
Einstein gravity is also exploited in condensed matter: some parameters such
as mass, spin, velocity, etc. are combined to provide an effective
"metric" that then is entered into the quantum mechanical equations
(e.g. Danilov et al. (1996) and Leonhard and Piwnicki (1999)).

Thus, the trend has been
forward the entire intricacy: the formalism of ψ-function penetrates to
the Planck length interior and the Einstein metric formalism advances to the
same scale as well. Nobody wishes accept the fact that on the size comparable
the de Broglie wavelength λ of an object methods of general relativity
fail. No one wants to go deeply into de Broglie's remark that the ψ-function
is only a reflection of some hidden variables of a particle moving in the real
physical space. The ψ-function is not the mother of particle nature and
therefore it cannot serve as a variable of the expansion of a particle's
characteristic in terms of ψ at the size less than the particle's de
Broglie wavelength λ.

2. New Understanding

Among new approaches
describing gravity in the microworld, we can notice the mathematical knot
theory (see, e.g. Pullin, 1993), which has been developed (Wallace, 2000)
attempting to find rules to establish when one knot can be transformed into
another without untying it. In the theory, the question is reduced to a
certain knot invariant problem, which does not change with knot deformations;
knot invariants being deformed constantly by gauge transformations should stay
unchangeable. The approach is similar in many aspects to concepts elaborated
in elementary particle physics.

Of special note is the
approach proposed by Bounias (1990, 2000) and Bounias and Bonaly (1994, 1996,
1997). Basing on the topology and the set theory, they have demonstrated that
the necessity of the existence of the empty set leads to the topological
spaces resulting in a "physical universe". Namely, they have
investigated links between physical existence, observability, and information.
The introduction of the empty hyperset has allowed a preliminary construction
of a formal structure that correlates with the degenerate cell of space
supporting conditions for the existence of a universe. Besides, among other
results we can point to their very promising hypothesis on a non-metric
topological distance as the symmetric difference between sets: this could be a
good alternative to the conventional metric distance which so far is still
treated as the major characteristic in all concepts employed in gravitational
physics, cosmology, and partly in quantum physics.

In my own line of
research I started from the fact that on the scale ~10-28 cm
constants of electromagnetic, weak and strong interactions as functions of
distance between interacting particles intersect (see, e.g., Okun, 1988). On
the other hand, in the high energy physics theorists deal with an abstract
"superparticle" which different states are electron, muon, quark,
etc. (see, e.g. Amaldi, 2000). A simple logical deduction suggests itself: the
physical space at the said range has a peculiarity that could be associated
with presence of structural blocks which one can call just superparticles (or
elementary cells, or balls). Then one may expect that a theory of the physical
space densely packed with those superparticles will be able to overcome many
difficulties which are insuperable in formal theories of both quantum gravity
and high energy physics. Thus a submicroscopic theory being based on the
structure of fine-grained space will be able to widely expand our knowledge
about the origin of matter, the foundations of quantum mechanics and the
foundations of quantum gravity.

The first step of the
theory (Krasnoholovets and Ivanovsky, 1993; Krasnoholovets, 1997, 2000a,
2000b) focused on the appearance of a particle from a superparticle, which
initially was found in the degenerate state. The particle has been defined as
a local curvature, or a local deformation of a superparticle and hence the
appearance of the deformation in a superparticle means the induction of mass
in it, m = CVsup/Vpart (C is the dimensional constant, Vsup
is the initial volume of a degenerate superparticle and Vpart is
the volume of the deformed superparticle, i.e. the volume of the created
particle). So the real space was regarded rather as a substrate, or quantum
aether, and the notion of a particle in it was adequately determined.

In condensed matter, we
meet the effect of the deformation of the crystal lattice in the surrounding
of a foreign particle and the solvation effect in liquids. Therefore, the
second step of the theory was the proposition that around a particle a
deformation coat was induced. This coat should play the role of a screen
shielding the particle from the degenerate space substrate. Within the coat,
the space substrate should be considered as a crystal and superparticles here
feature mass. Thus the coat may be treated as a peculiar crystallite. The size
of the crystallite was associated with the Compton wavelength of the particle,
λCom = h/mc.

The next step needed a
correct physical model of the motion of the particle. From the solid state
physics we know that the motion of particles is accompanied with the motion of
elementary excitations of some sort, namely, the particle when is moving in a
solid emits and absorbs quasi-particles such as excitons, phonons, etc. By
analogy, the motion of the physical "point" (particle cell) in the
entirely packed space must be accompanied by the interaction with coming
"points" of the space, i.e. superparticle cells. Hence the particle
is scattered by structural blocks of the space that in turn should lead to the
induction of elementary excitations in superparticles, which contact the
moving particle. The corresponding excitations were called "inertons"
as the notion "inertia" means the resistance to the motion (thus
particle's inertons reflect resistance on the side of the space in respect to
the moving particle). Each inerton carries a bit of the particle deformation,
that is, an inerton is characterized by the mass as well. An inerton migrates
from superparticle to superparticle by relay mechanism. The deformation coat,
or crystallite, is pulled by the particle: superparticles, which form the
crystallite, do not move from their positions in the space substrate, however,
the massive state of crystallite's superparticles is passed on from
superparticles to superparticles along the whole particle path.

3. Submicroscopic
Mechanics

The Lagrangian that is
able to satisfy the described motion of a particle and the ensemble of its
inertons can be written as (Krasnoholovets and Ivanovsky, 1993)

where the first term
characterizes the kinetics energy of the particle, the second term
characterizes the kinetics energy of the ensemble of N inertons, emitted from
the particle and the third term specifies the contact interaction between the
particle and its inertons. Xi is the ith component of the position
of the particle; gij is metric tensor components generated by the
particle; (v0)i is the ith component of the initial
particle's velocity vector v0. Index s corresponds to the number of
respective inertons; x(s)j is the component of the
position of the sth inerton; ĝ(s)ij is the metric tensor
components of the position of the sth inerton. 1/T(s) is the
frequency of collisions of the particle with the sth inerton. Kronecker's
symbol δt-∆t(s), t(s) provides the agreement of proper
times of the particle t and the sth inerton t(s) at the instant of
their collision (∆t(s) is the time interval after expire of
which, measuring from the initial moment t = 0, the moving particle emits the
sth inerton). The interaction operator Ö {giqθĝ(s)qj}
possesses special properties: θ = 0 during a short time interval δt
when the particle and the sth inerton is in direct contact and θ = 1 when
the particle and the sth inerton fly apart along their own paths. Note that in
the model presented the metric tensor characterizes changing in sizes of the
particle and superparticles.

In the so-called
relativistic case when the initial velocity v0 of the particle is
close to the speed of light c, the relativistic mechanics prescribes the
Lagrangian

Lrel = –M0c2
Ö {1–v02/c2}. (3)

On examination of the
relativistic particle, we shall introduce into the Lagrangian (3) terms, which
describe inertons and their interaction with the particle. For this purpose,
the following transformation in (3) should be made (Krasnoholovets, 1997)

written for the particle
(Q = Xi) and the sth inerton (Q = x(s)i)
coincide for the Lagrangians L =L, (3), and L =Lrel,
(4). This is true only (Dubrovin et al. (1986)) in the case when the time t
entered into the Lagrangians (3) and (4) is considered as the natural
parameter, i.e. t = l/v0 where l is the length of the particle
path.

For the variables X(s)k
≡ Xk(t(s)) and x(s)k
≡ xk(t(s)) one obtains from eq. (5) the equations
of extremals (written as functions of the proper time t(s) of the
emitted sth inerton):

Eq. (8) specifies the
merging the particle and the sth inerton into a common system. This means the
acceleration that the particle experiences, coincides with that of the sth
inerton. Then the difference in the first set of parentheses in eq. (8) is
equal to zero and instead of eq. (8) we get

Coefficients Γkij
and Γk(s)ij are generated by the particle
mass M and the sth inerton mass m(s), respectively, and that is why
Γkij /Γk(s)ij =
M/m(s). This signifies that relationship (9) can be rewritten
explicitly

Mv0s2
= m(s)c2 (10)

for diagonal metric
components of the particle and inerton velocities, (v0(s) is the
velocity of the particle after its scattering by the sth inerton with initial
velocity c).

When the particle and the
sth inerton bounce apart, we must solve the total equations of motion, (6) and
(7), i.e., all terms in the equations should be held. However, if we allow the
metric tensors are constant, the equations of motion may be simplified to the
form that does not include the second nonlinear term in both eqs. (6) and (7).
The structure and properties of the metric tensors can be chosen as follow

gij= δij
M; gij= δij/M; gki giq =
δqk;

ĝ(s)ij=
δij m(s); ĝ(s)ij= δij/m(s);
ĝ(s)ki ĝ(s)iq = δqk.
(11)

Thus having given gij
and ĝ(s)ij are equal to constant, the second term in both eqs.
(7) and (8) is made to be reduced to zero. Relationships (11) and (10) allow
transforming of the interaction operator in eqs. (7) and (8) to forms

gki (giqθĝ(s)qj)1/2
→ Ö {m(s)/M} = v0(s)k/c; (12)

ĝki (giqθĝ(s)qj)1/2
→ Ö {M/m(s)}= c /v0(s)k
(13)

where v0(s)k
is the kth component of the vector v0(s). Thus expressions (12) and
(13) permit the transformation of eqs. (7) and (8) (in which second terms are
dropped) to the form

If we consider the
ensemble of inertons as the whole object, an inerton cloud with the rest mass
m0, which surrounds a moving particle with the rest mass M0
then the Lagrangian may be presented as

L = – M0c2
{1– (1/M0c2)[M0 (dX/dt)2 + m0
(dx/dt)2

– (2π/T) Ö {M0m0}
(X dx/dt + v0x)]}1/2. (16)

Thus the particle moves
along the X-axis with the velocity dX/dt (v0 is the initial
velocity); x is the distance between the inerton cloud and the particle, dx/dt
is the velocity of the inertons cloud, and 1/T is the frequency of collisions
between the particle and cloud. The equations of motion are reduced to the
following

d2 X/dt2 +
(πv0/cT) dx/dt = 0; (17)

d2 x/dt2 –
(πc /v0T) (dX/dt – v0)= 0. (18)

The corresponding
solutions to eqs. (17) and (18) for the particle and the inerton cloud are

dX/dt= v0
(1– |sin (πt/T)| );

X(t) = v0t + (λ/π)
{(-1)[t/T] cos(πt/T) – (1 + 2 [t/T])};

λ = v0T;
(19)

x = (Λ/π) |sin
(πt/T)|;

dx/dt = c(– 1)[t/T]
cos (πt/T);

Λ = cT. (20)

Expressions (19) show
that the velocity of the particle periodically oscillates and λ is the
amplitude of particle's oscillations along its path. In particular, λ is
the period of oscillation of the particle velocity that periodically changes
between v0 and zero. The inertons cloud periodically leaves the
particle and then comes back; Λ is the amplitude of oscillations of the
cloud.

The frequency of
collisions of the particle with the inerton cloud allows the presentation in
two ways: 1) via the collision of the particle with the cloud, i.e., 1/T = v0/λ
and 2) via the collision of the inerton cloud with the particle, i.e., 1/T =
c/Λ. These two expressions result into the relationship

v0/λ = c/Λ,
(21)

which connects the
spatial period λ of oscillations of the particle with the amplitude
Λ of the inertons cloud, i.e., maximal distance to which inertons are
removed from the particle.

If we introduce a new
variable

dκ/dt = dx/dt – (π/T)
X Ö {M0/m0} (22)

in the Lagrangian (16),
we arrive to the canonical form on variables for the particle

L = – M0c2{1–
(1/M0c2) [M0 (dX/dt)2 – M0
(2π/2T)2X2

+ m0 (dκ/dt)2
– (2π/T) v0x Ö {M0/m0}]}1/2.
(23)

This Lagrangian allow us
to obtain (Krasnoholovets, 1997) the effective Hamiltonian of the particle
that describes its behavior relative to the center of inertia of the particle-inerton
cloud system

Heff = p2/2M
+ M (2π/2T)2 X2/2 (24)

where M = M0/Ö
{1–v02/c2} (and also m = m0/Ö
{1–v02/c2}). The harmonic oscillator
Hamiltonian (24) allows one to write the Hamilton-Jacobi equation for a
shortened action S1 of the particle

(∂S1/∂X)2/2M
+ M (2π/2T)2 X2/2 = E. (25)

Here E is the energy of
the moving particle. Introduction of the action-angle variables leads to the
following increment of the particle action within the cyclic period 2T (Krasnoholovets
and Ivanovsky, 1993)

∆S1 =
" p dX = E.2T. (26)

Eq. (26) one can write
via the frequency ν = 1/2T as well. At the same time 1/T is the frequency
of collisions of the particle with its inertons cloud. Owing to the relation E
= Mv02 /2 we also get

∆S1 = Mv0.v0T
= p0λ (27)

where p0 = Mv0
is the particle initial momentum. Now if we equate the values ∆S1
and Planck's constant h, we obtain instead of expressions (26) and (27) major
relationships 1, which form the basis of conventional quantum mechanics.

De Broglie (1986), when
writing relationships (1), noted that they resulted from the comparison of the
action of a particle moving rectilinearly and uniformly (with the energy E and
the momentum Mv0) and the phase of a plane monochromatic wave
extended in the same direction (with the frequency E/h and the wavelength h/Mv0).
Yet the first relation in (1) he considered as the main original axiom of
quantum theory.

In our case, expressions
(26) and (27) have been derived starting from the Hamiltonian (24) or the
Hamilton-Jacobi eq. (25) of the particle. The main peculiarity of our model is
that the Hamiltonian and the Hamilton-Jacobi equation describe a particle
whose motion is not uniform but oscillatory. It is the space substrate, which
induces the harmonic potential responding to the disturbance of the space by
the moving particle. The oscillatory motion of the particle is characterized
by the relation

λ = v0T
(28)

which connects the
initial velocity of the particle v0 with the spatial period of
particle oscillations λ (or the free path length of the particle), and
the time interval T during which the particle remains free, i.e. does not
collide with its inerton cloud. On the other hand, relation (28) holds for a
monochromatic plane wave that spreads in the real physical space: λ is
the wavelength, T is the period and v0 is the phase velocity of the
wave. Thus with the availability of the harmonic potential, the behavior of
the particle follows the behavior of a wave and, therefore, such a motion
should be marked by a very specific value of the adiabatic invariant, or
increment of the particle action ∆S1 within the cyclic
period. It is quite reasonable to assume that in this case the value of
∆S1 is minimum, which is equal to Planck's constant h.

Two relationships (1)
immediately allow the deduction of the Schrödinger equation (de Broglie,
1986). Moreover, the presence of the proper time of a particle in the
Schrödinger equation (Krasnoholovets, 1997) signifies that the equation is
Lorentz invariant. The wave ψ-function acquires a sense of the imaging a
real wave function that characterizes the motion of a complicated formation
– the particle and its inerton cloud. The real wave function (and its wave
ψ-function imaging or map) is defined in the range that is exemplified by
the dimensions of the particle's inerton cloud: λ along the particle path
and 2Λ in transversal directions. In such a manner, inertons acquire the
sense of a substructure of the matter waves and should be treated as carriers
of inert properties of matter. Heisenberg's uncertainty relations gain a
deterministic interpretation as a quantum system now is complemented by the
inerton cloud; therefore, an unknown value of the momentum of the particle
automatically is compensated by the corresponding momentum of the particle's
inertons.

4. Spin and Relativistic
Approximation

The notion of spin of a
particle is associated with an intrinsic particle motion. Several tens of
works have been devoted to the spin problem. Major of them is reviewed in the
recent author's paper (Krasnoholovets, 2000a). Here we add some recent
references (Chashihin, 2000; Rangelov, 2001; Danilov et al., 1996: Plyuschay,
1989, 1990, 2000). Main ideas of the works quoted in (Krasnoholovets, 2000a)
and in the mentioned references are reduced to a moving particle that is
surrounded by a wave, or a small massless particle, or an ensemble of small
massless particles, which engage in a circular motion.

Having tried the
introduction of the notion of spin in the concept presented, let us look at
the situations in which the particle spin manifest itself explicitly. First,
it appears as an additional member ± ħ/2 to the projection onto the
z-axis of the moment of momentum r´ Mv0 of a
particle. Second, it introduces the correction ± eBzħ/(2M) to
the energy of a charged particle in the magnetic field with the projection of
the induction onto the z-axis equals to Bz. Third, it provides for
the Pauli exclusion principle.

Of course, it seems quite
reasonable to assume that the spin in fact reflects some kind of proper
rotation of the particle. However, we should keep in mind that the operation
'rotor' is typical for the electromagnetic field that the particle generates
in the environment when starts to move. In other words, the appearance of the
electromagnetic field in the particle surrounding one may associate just with
its proper rotation of some sort. In our concept, superparticles that form the
space net are not rigid; they fluctuate and allow local stable and unstable
deformations. Thus the particle may be considered as not rigid as well. In
this case along with an oscillating rectilinear motion, the particle is able
to undergo some kind of an inner pulsation, like a drop. Besides the pulsation
can be oriented either along the particle velocity vector or diametrically
opposite to it. Then the Lagrangians (16) and (23) change to the matrix form (Krasnoholovets,
2000a)

L = || Lα
||, α = ↑, ↓. (29)

The function Lα
can be written as

Lα
= – gc2{1 – [U(spat) + U(intr) α ] /
gc2}1/2 . (30)

Here U(spat)
is the same as in expression (2) and U(intr) α is similar to U(spat),
however, all spatial coordinates (and velocities) are replaced for the
intrinsic ones: X → Ξα for the
particle and x → ξα for the inerton
cloud. So inertons carry bits of the particle pulsation as well. The intrinsic
motion is treated as a function of the proper time of the particle t. Then the
equations of motion and the solutions to them are quite similar to those
obtained in the previous section. The intrinsic velocity dΞα/dt
ranges between ± v0 ("+" if α = ↑ and "–"
if α = ↓) and zero; dξα/dt ranges between ± c
and zero within the segment 2Λ of the spatial path of the inerton cloud.
Such a motion is characterized by relationships similar to (26) and (27) and
hence is marked by Planck's constant h.

The intrinsic variables
do not appear in the case of a free moving particle. However, an external
field being superimposed on the system is able to engage into the variables.
Then we can write the wave equation for the spin variable of the particle

(Πα2
/ 2M – eαεα ) χα =
0 (31)

where the operator Πα
= (πα – eA) and πα=
– iħd/dΞα is the operator of the intrinsic
momentum of the particle, e and A arethe electric charge and
the vector potential of the field, respectively. χα is
the eigenfunction and εα is theeigenvalue;
the function eα = 1 if α = ↑ and eα
= – 1 if α = ↓.

If the induction of the
magnetic field has only one component Bz aligned with the z-axis,
the solution to eq. (31) becomes

εα =
eαeħBz / 2M; (32)

χα =
π-1/4 exp[– (πα x
– eAx) / 2eħBz]. (33)

So ε↑,↓
= ± eħBz/2M and therefore the eigenvalues of the so-called
spin operator S↑↓ are

S↑,↓z
= ± ħ/2; S↑,↓x = S↑,↓y =
0. (34)

Thus, the intrinsic
motion introduced above satisfies the behavior of a particle in the magnetic
field. The total orbital moment of the electron in an atom includes the spin
contribution proceeding just from the interaction of the electron with an
magnetic field. Moreover, the availability of two possible antipodal intrinsic
motions of the particle allows the satisfaction of the Pauli exclusion
principle. Consequently, the model of the spin described complies with the
three said requirements.

Now the total Hamiltonian
of a particle can be represented in the form of

H↑,↓
= c Ö {p2 + π↑,↓2
+ M02c2} (35)

(a similar Hamiltonian
describes the particle's inerton cloud). As can easily be seen from expression
(35), the spin introduces an additional energy to the particle Hamiltonian
transforming it to a matrix form. Then following Dirac we can linearize the
matrix H↑,↓ anddoing so we will arrive to
the Dirac Hamiltonian

HDirac = cαp
+ ρ3M0c2. (36)

At this point,
information on the matric operators π↑,↓ goes
into the Dirac matrices. Thus from the physical point of view the Dirac
transformation (36) is substantiated only in the case when the initial
Hamiltonian is a matrix as well. And just this fact has been demonstrated in
the theory proposed.

The Dirac formalism is
correct in the range r ≥ h/Mc and is restricted by the amplitude of
inerton cloud Λ= λc/v0. At r < h/Mc the approach
described above can easily be applied. It has been pointed out (Krasnoholovets,
2000a) that the inerton cloud and the oscillatory mode of the crystallite's
superparticles, which vibrate in the environment of the particle, cause the
nature of spinor components. Two possible projections of spin enlarge the
total number of the Dirac matrices and the spinors to four.

The submicroscopic
consideration allows one sheds light on the interpretation of the so-called
negative kinetic energy and the negative mass of rest of a free particle,
which enter into the solutions of the corresponding Dirac equation (see, e.g.
Schiff, 1959). The negative spectral eigenvalues E_= –Ö {c2p2
+ m2c4} are interpreted as states with the negative
energy of the particle (and because of that Dirac proposed to refer it to the
energy of the positron). However, the presence of the inerton cloud that
oscillates near the particle lets us to construe the eigenvalues of the
particle as a spectrum of "left" and "right" inerton waves
which respectfully emitted and absorbed by the particle. Such waves, φα
_ = φα(r – at) and φα + =
φα(r + at) where α specifies the spin projection,
depend on the space variables identically, but the time variable t is entered
as either +t or –t. In quantum mechanics the operator iħ∂/∂t
just corresponds to the particle energy E. Thus, we can interpret the positive
eigenvalue E+ as the total energy of the inerton cloud that moves
away from the particle while the negative eigenvalue E_ as the total energy of
the inerton cloud that comes back to the particle.

It is interesting to note
that some parallels we can meet in the recent research conducted by Dubois
(1999a,b), who has studied anticipation in physical systems considering
anticipation as their inner property which is embedded in the system but is
not a model-based predicted. In particular, it has been shown by Dubois that
such a property is inherent to electromagnetism and quantum mechanics. Namely,
Dubois (1999b) started from space-time complex continuous derivatives which
were constructed in such a way that gave the discrete forward and backward
derivatives ∂± /∂t. Dubois's methodology may be justified in
terms of the present submicroscopic approach because the derivative ∂–/∂t
could be referred to inertons flying away from the particle and the derivative
∂+/∂t could be assigned to inertons moving backward to
the particle. Besides the two types of velocities are present in anticipatory
physical systems, so called "phase" and "group"
velocities. These two velocities would also be ascribed to two opposite flows
of inertons. We can also emphasize that the Dubois' idea about the masses of
particles as properties of space-time shifts is also very close to the
author's hypothesis on mass as a local deformation in the space net. Note that
the hypothesis has found future trends (Bounias and Krasnoholovets, 2002): it
allows evidence in terms of the topology and fractal geometry.

5. Inertons in Action:
Experimental Verification

§1. The
photoelectric effect occurring under strong irradiation in the case that the
energy of the incident light is essentially smaller than the ionization
potential of gas atoms and the work function of the metal has been
reconsidered from the submicroscopic viewpoint. It has been shown (Krasnoholovets,
2001a) that the (nonlinear) multiphoton theory, which has widely been used so
far, and the effective photon concept should be changed to a new methodology.
The author's approach was based on the hypothesis that inerton clouds are
expanded around atoms' electrons. That means that the effective cross-section
σ of an atom's electron together with the electron's inerton cloud falls
within the range between λ2 and Λ2 (i.e. 10-16
cm2 < σ < 10-12 cm2) that much
exceeds the cross-section area of the actual atom size, 10-16 cm2.
The intensity of light in focused laser pulses used for the study of gas
ionization and photoemission from metals was of the order of 1012
to 1015 W/cm2. Thus several tens of photons
simultaneously should pierce the electron's inerton cloud and at least several
of them could be engaged with the cloud's inertons and scattered by them.
Consequently, the electron receives the energy needed to release from an atom
or metal. The theory indeed has been successfully applied to the numerous
experiments (Krasnoholovets, 2001a).

§2. In
condensed media, inerton clouds of separate particles (electrons, atoms, and
molecules) should overlap forming the entire elastic inerton field, which
densely floods in the media. It has been theoretically shown (Krasnoholovets
and Byckov, 2000) that in this case the force matrix W that determines
branches of acoustic vibrations in solids comprises of two members: W = Vac
+ Viner. Here the first member is responsible for the usual
elastic electromagnetic interaction of atoms and is responsible just for the
availability of acoustic properties of solids, but the second one is
originated from the overlapping of atoms' inerton clouds. It is remarkable
that each of the members has the same right. Therefore, an inerton wave
striking an object will influence the object much as an applied ultrasound.
Among the features of ultrasound, one can call destroying, polishing, and
crushing. It was anticipated that inerton waves would act on specimens in a
similar manner. A power source of inerton waves is the Earth: any mechanical
fluctuations in the Earth should generate corresponding inerton waves. Two
types of inerton flows one can set off in the terrestrial globe. The first
flow is caused by the proper rotation of the Earth. Let A be a point on the
Earth surface from which an inerton wave is radiated. If the inerton wave
travels around the globe along the West-East line, its front will pass a
distance L1 = 2πREarth per circle. The second flow
spreads along the terrestrial diameter; such inerton waves radiated from A
will come back passing distance L2 = 4REarth.The
ratio is

L1/L2
= π /2. (37)

If in point A we locate a
material object which linear sizes (along the West-East line and perpendicular
to the Earth surface) satisfy relation (37), we will receive a resonator of
the Earth inerton waves.

We have studied specimens
(razor blades) put into the resonator for several weeks. By using the scanning
electron microscope, in fact, we have established difference in the fine
morphological structure of cutting edge of the razor blades while the
morphologically more course structure remains well preserved.

Note that the Earth
inerton field is also the principal mover that launched rather fantastic
quantum chemical physical processes in Egypt pyramids (Krasnoholovets, 2001b),
power plants of the ancients that has recently been proved by Dunn (1998).

§3. Just
recently, the inerton concept has been justified in the experiment on the
searching for hydrogen atoms clustering in the δ-KIO3·HIO3
crystal (Krasnoholovets et al., 2001). It has been assumed that vibrating
atomsshould induce the inerton field within the crystal. This in turn
should change the paired potential of interatomic interaction. Taking into
account such a possible alteration in the potential, we have calculated the
number of hydrogen atoms in a cluster and predicted its properties. Then the
crystal has been investigated by using the Bruker FT IR spectrometer in the
400 to 4000 cm-1 spectral range. Features observed in the spectra
unambiguously have been interpreted just as clustering of hydrogen atoms.

6. Concluding Remarks

Thus, we have uncovered
that the interpretation of quantum mechanics in the framework of the double
solution theory indeed is possible. However, the theory presented is
distinguished from de Broglie's (1987), which he actively developed seeking
for the solution of deterministic interpretation of the problem. The major
point of the given concept is an original cellular construction of a real
space, the introduction of notions of the particle, mass, and elementary
excitations of the space. The mechanics constructed is based on the
Lagrangians (16) and (23), equations of motion, and solutions to them,
(17)-(22). The Lagrangians explicitly include elementary excitations of the
space, which accompany a moving particle and directly interact with the
particle. The main peculiarities of the mechanics called submicroscopic
quantum (or wave) mechanics are the free path lengths for the particle λ
and its inerton cloud Λ and, because of that, the mechanics is similar to
the kinetics theory. The particle velocity v0 is connected with
λ by relation v0 = λ/T where 1/T is the frequency of the
particle collisions with the inerton cloud (and 1/2T = ν is the frequency
of the particle oscillation along its path). Since the motion of the particle
is of oscillating nature, it permits the construction of the Hamiltonian-Jacobi
equation (25) and the obtaining the minimum increment of the particle action
within the period ν-1 that is identified with Planck's
constant h. This allows one derives the principal quantum mechanical relations
(1) and then constructs the Schrödinger and Dirac formalisms.

Submicroscopic quantum
mechanics has solved the spin problem reducing it to special intrinsic
pulsations of a moving particle. As a result, an additional correction
(positive or negative) is introduced to the particle's Hamiltonian
transforming it to a matrix form that in its turn has provided the reliable
background to the Dirac's linearization of the classical relativistic
Hamiltonian.

Inertons are treated as a
substructure of the matter waves and yet inertons surrounding moving particles
are identified with carriers of inert properties of the particles. The inerton
concept also determines the boundaries of employment of the wave ψ-function
and spinor formalisms reducing the boundaries to the range covered by inerton
cloud amplitude Λ of the particle studied.

At last, inertons, which
widely manifest themselves in numerous experiments, can be treated as a basis
for anticipation in physical systems because just inertons represent those
inner properties to which Dubois (1999a,b) referred constructing anticipation
as actually embedded in the systems.

Further studies need
widening the scope of applying of quantum mechanics. In particular, one could
apply inertons to the problem of quantum gravity because inertons may also be
considered as real carriers of gravitational interaction. Bounias (2001) has
just found other application of inertons, namely, to biological systems: the
availability of the inerton wave function of an object allowed him to
construct the Hamiltonian of living organism considering it as an anticipatory
operator of evolution.

Bounias M. (2000). The
Theory of Something: a Theorem Supporting the Conditions for Existence of a
Physical Universe, from the Empty Set to the Biological Self, International
Journal of Computing Anticipatory Systems5-6, 1-14.

Bounias M. (2001).
"The Hamiltonian of Living Organisms: an Anticipatory Operator of
Evolution." Computing Anticipatory Systems: CASYS'01 – Fifth
International Conference. Edited by Daniel M. Dubois, Published by The
American Institute of Physics, AIP Conference Proceedings, to be published.